%C This is also the number of moves that it takes n frogs to swap places with n toads on a strip of 2n + 1 squares (or positions, or lily pads) where a move is a single slide or jump, illustrated for n = 2, a(n) = 8 by

%C T T - F F

%C T - T F F

%C T F T - F

%C T F T F -

%C T F - F T

%C - F T F T

%C F - T F T

%C F F T - T

%C F F - T T

%C I was alerted to this by the Holton article, but on consulting Singmaster's sources, I find that the puzzle goes back at least to 1867.

%C Probably the first to publish the number of moves for n of each animal was Edouard Lucas in 1883. (End)

%C Using four consecutive triangular numbers t1, t2, t3 and t4, plot the points (0, 0), (t1, t2), and (t3, t4) to create a triangle. Twice the area of this triangle are the numbers in this sequence beginning with n = 1 to give 8. - _J. M. Bergot_, May 03 2012

%C Given a particle with spin S = n/2 (always a half-integer value), the quantum-mechanical expectation value of the square of the magnitude of its spin vector evaluates to <S^2> = S(S+1) = n(n+2)/4, i.e., one quarter of a(n) with n = 2S. This plays an important role in the theory of magnetism and magnetic resonance. - _Stanislav Sykora_, May 26 2012

%C Also for n>0, a(n) is the number of times that n-1 occurs among the first (n+1)! terms of A055881. - _R. J. Cano_, Dec 21 2016

%C The second diagonal of composites (the only prime is number 3) from the right on the Klauber triangle (see Kusniec link), which is formed by taking the positive integers and taking the first 1, the next 3, the following 5, and so on, each centered below the last. - _Charles Kusniec_, Jul 03 2017

%C Also the number of independent vertex sets in the n-barbell graph. - _Eric W. Weisstein_, Aug 16 2017